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Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_linsys_complex_posdef_tridiag_solve (f04cg)

## Purpose

nag_linsys_complex_posdef_tridiag_solve (f04cg) computes the solution to a complex system of linear equations AX = B$AX=B$, where A$A$ is an n$n$ by n$n$ Hermitian positive definite tridiagonal matrix and X$X$ and B$B$ are n$n$ by r$r$ matrices. An estimate of the condition number of A$A$ and an error bound for the computed solution are also returned.

## Syntax

[d, e, b, rcond, errbnd, ifail] = f04cg(d, e, b, 'n', n, 'nrhs_p', nrhs_p)
[d, e, b, rcond, errbnd, ifail] = nag_linsys_complex_posdef_tridiag_solve(d, e, b, 'n', n, 'nrhs_p', nrhs_p)

## Description

A$A$ is factorized as A = LDLH$A=LD{L}^{\mathrm{H}}$, where L$L$ is a unit lower bidiagonal matrix and D$D$ is a real diagonal matrix, and the factored form of A$A$ is then used to solve the system of equations.

## References

Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia http://www.netlib.org/lapack/lug
Higham N J (2002) Accuracy and Stability of Numerical Algorithms (2nd Edition) SIAM, Philadelphia

## Parameters

### Compulsory Input Parameters

1:     d( : $:$) – double array
Note: the dimension of the array d must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
Must contain the n$n$ diagonal elements of the tridiagonal matrix A$A$.
2:     e( : $:$) – complex array
Note: the dimension of the array e must be at least max (1,n1)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$.
Must contain the (n1)$\left(n-1\right)$ subdiagonal elements of the tridiagonal matrix A$A$.
3:     b(ldb, : $:$) – complex array
The first dimension of the array b must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The second dimension of the array must be at least max (1,nrhs)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$
The n$n$ by r$r$ matrix of right-hand sides B$B$.

### Optional Input Parameters

1:     n – int64int32nag_int scalar
Default: The first dimension of the array b.
The number of linear equations n$n$, i.e., the order of the matrix A$A$.
Constraint: n0${\mathbf{n}}\ge 0$.
2:     nrhs_p – int64int32nag_int scalar
Default: The second dimension of the array b.
The number of right-hand sides r$r$, i.e., the number of columns of the matrix B$B$.
Constraint: nrhs0${\mathbf{nrhs}}\ge 0$.

ldb

### Output Parameters

1:     d( : $:$) – double array
Note: the dimension of the array d must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
If ${\mathbf{ifail}}={\mathbf{0}}$ or n + 1$\mathbf{n}+{\mathbf{1}}$, d stores the n$n$ diagonal elements of the diagonal matrix D$D$ from the LDLH$LD{L}^{\mathrm{H}}$ factorization of A$A$.
2:     e( : $:$) – complex array
Note: the dimension of the array e must be at least max (1,n1)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$.
If ${\mathbf{ifail}}={\mathbf{0}}$ or n + 1$\mathbf{n}+{\mathbf{1}}$, e stores the (n1)$\left(n-1\right)$ subdiagonal elements of the unit lower bidiagonal matrix L$L$ from the LDLH$LD{L}^{\mathrm{H}}$ factorization of A$A$. (e can also be regarded as the conjugate of the superdiagonal of the unit upper bidiagonal factor U$U$ from the UHDU${U}^{\mathrm{H}}DU$ factorization of A$A$.)
3:     b(ldb, : $:$) – complex array
The first dimension of the array b will be max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The second dimension of the array will be max (1,nrhs)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$
ldbmax (1,n)$\mathit{ldb}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
If ${\mathbf{ifail}}={\mathbf{0}}$ or n + 1$\mathbf{n}+{\mathbf{1}}$, the n$n$ by r$r$ solution matrix X$X$.
4:     rcond – double scalar
If ${\mathbf{ifail}}={\mathbf{0}}$ or n + 1$\mathbf{n}+{\mathbf{1}}$, an estimate of the reciprocal of the condition number of the matrix A$A$, computed as rcond = 1 / (A1,A11)${\mathbf{rcond}}=1/\left({‖A‖}_{1},{‖{A}^{-1}‖}_{1}\right)$.
5:     errbnd – double scalar
If ${\mathbf{ifail}}={\mathbf{0}}$ or n + 1$\mathbf{n}+{\mathbf{1}}$, an estimate of the forward error bound for a computed solution $\stackrel{^}{x}$, such that x1 / x1errbnd${‖\stackrel{^}{x}-x‖}_{1}/{‖x‖}_{1}\le {\mathbf{errbnd}}$, where $\stackrel{^}{x}$ is a column of the computed solution returned in the array b and x$x$ is the corresponding column of the exact solution X$X$. If rcond is less than machine precision, then errbnd is returned as unity.
6:     ifail – int64int32nag_int scalar
${\mathrm{ifail}}={\mathbf{0}}$ unless the function detects an error (see [Error Indicators and Warnings]).

## Error Indicators and Warnings

Errors or warnings detected by the function:

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

ifail < 0andifail999${\mathbf{ifail}}<0 \text{and} {\mathbf{ifail}}\ne -999$
If ifail = i${\mathbf{ifail}}=-i$, the i$i$th argument had an illegal value.
ifail = 999${\mathbf{ifail}}=-999$
Allocation of memory failed. The double allocatable memory required is n. In this case the factorization and the solution X$X$ have been computed, but rcond and errbnd have not been computed.
ifail > 0andifailN${\mathbf{ifail}}>0 \text{and} {\mathbf{ifail}}\le {\mathbf{N}}$
If ifail = i${\mathbf{ifail}}=i$, the leading minor of order i$i$ of A$A$ is not positive definite. The factorization could not be completed, and the solution has not been computed.
W ifail = N + 1${\mathbf{ifail}}={\mathbf{N}}+1$
rcond is less than machine precision, so that the matrix A$A$ is numerically singular. A solution to the equations AX = B$AX=B$ has nevertheless been computed.

## Accuracy

The computed solution for a single right-hand side, $\stackrel{^}{x}$, satisfies an equation of the form
 (A + E) x̂ = b, $(A+E) x^=b,$
where
 ‖E‖1 = O(ε) ‖A‖1 $‖E‖1=O(ε) ‖A‖1$
and ε$\epsilon$ is the machine precision. An approximate error bound for the computed solution is given by
 (‖x̂ − x‖1)/(‖x‖1) ≤ κ(A) (‖E‖1)/(‖A‖1) , $‖x^-x‖1 ‖x‖1 ≤ κ(A) ‖E‖1 ‖A‖1 ,$
where κ(A) = A11A1$\kappa \left(A\right)={‖{A}^{-1}‖}_{1}{‖A‖}_{1}$, the condition number of A$A$ with respect to the solution of the linear equations. nag_linsys_complex_posdef_tridiag_solve (f04cg) uses the approximation E1 = εA1${‖E‖}_{1}=\epsilon {‖A‖}_{1}$ to estimate errbnd. See Section 4.4 of Anderson et al. (1999) for further details.

The total number of floating point operations required to solve the equations AX = B$AX=B$ is proportional to nr$nr$. The condition number estimation requires O(n)$\mathit{O}\left(n\right)$ floating point operations.
See Section 15.3 of Higham (2002) for further details on computing the condition number of tridiagonal matrices.
The real analogue of nag_linsys_complex_posdef_tridiag_solve (f04cg) is nag_linsys_real_posdef_tridiag_solve (f04bg).

## Example

function nag_linsys_complex_posdef_tridiag_solve_example
d = [16;
41;
46;
21];
e = [ 16 + 16i;
18 - 9i;
1 - 4i];
b = [ 64 + 16i,  -16 - 32i;
93 + 62i,  61 - 66i;
78 - 80i,  71 - 74i;
14 - 27i,  35 + 15i];
[dOut, eOut, bOut, rcond, errbnd, ifail] = nag_linsys_complex_posdef_tridiag_solve(d, e, b)

dOut =

16
9
1
4

eOut =

1.0000 + 1.0000i
2.0000 - 1.0000i
1.0000 - 4.0000i

bOut =

2.0000 + 1.0000i  -3.0000 - 2.0000i
1.0000 + 1.0000i   1.0000 + 1.0000i
1.0000 - 2.0000i   1.0000 - 2.0000i
1.0000 - 1.0000i   2.0000 + 1.0000i

rcond =

1.0862e-04

errbnd =

1.0221e-12

ifail =

0

function f04cg_example
d = [16;
41;
46;
21];
e = [ 16 + 16i;
18 - 9i;
1 - 4i];
b = [ 64 + 16i,  -16 - 32i;
93 + 62i,  61 - 66i;
78 - 80i,  71 - 74i;
14 - 27i,  35 + 15i];
[dOut, eOut, bOut, rcond, errbnd, ifail] = f04cg(d, e, b)

dOut =

16
9
1
4

eOut =

1.0000 + 1.0000i
2.0000 - 1.0000i
1.0000 - 4.0000i

bOut =

2.0000 + 1.0000i  -3.0000 - 2.0000i
1.0000 + 1.0000i   1.0000 + 1.0000i
1.0000 - 2.0000i   1.0000 - 2.0000i
1.0000 - 1.0000i   2.0000 + 1.0000i

rcond =

1.0862e-04

errbnd =

1.0221e-12

ifail =

0